DAGs, d-separation, paths of two vertices: confusion

Question

I'm taking a class on Bayesian probability. Briefly, my question is about directed acyclical graphs (DAGs) and d-separation of parent and child vertices, conditional on the parent. (This question arose while checking whether a DAG satisfies the Parental Markov Condition.)

Background

Some terminology and definitions taken from Pearl (2013) (feel free to skip).

By path we mean a sequence of consecutive edges (of any directionality) in a graph.

Def. 1 (d-separation): Vertices in the set $A$ are d-separated by vertices in set $B$ if and ony if either (1) or (2) is satisfied:

(1) There exists a path $x \to y \leftarrow z$, such that
$\hbox{ }$(1.1) the vertex $y \not\in B$, and
$\hbox{ }$(1.2.) for all $y^\prime$, such that $y^\prime$ is a descendant of $y$, we have $y^\prime \not\in B$.

(2) There exists a path $x \to y \to z$, or alternatively $x \leftarrow y \leftarrow z$, or also $x \leftarrow y \to z$ such that
$\hbox{ }$(2.1) the vertex $y \in B$.

Question.

Now, consider the following two vertice graph, with vertices $\{a, b\}$ and directed edges $\{(b, a)\}$.

The picture of the DAG is here. Sadly, not allowed to put it in the post. The DAG just looks like this: $$b \longrightarrow a$$

My question: Is $a$ d-separated from $b$, given $b$ in the two vertice directed acyclical graph? And why?

Let's see. We let $A = \{b, a\}$ and the so-called conditional set $B = \{b\}$. It seems to me that conditions (1) and (2) of the above Definition 1 fail, because there is no three-vertice path in the DAG above.

However, if one is allowed to go, so to speak, back and forth between vertices, the following path would satisfy condition (2) of Definition 1: $$a \leftarrow b \to a$$ because $b$ is a fork and $b \in \{b\} = B$. However, in a path, all vertices and edges are supposed to be distinct (Bender & Williamson 2010, p. 162.), which is not the case displayed above—so it shouldn't be a path. However, I didn't find this restriction in Pearl (2010). As you can see, I'm going in circles. What am I misunderstanding about d-separation?

Answer 1

I am pretty sure $x$ and $z$ are supposed to be distinct.

The notion of $d$-separation captures the idea of "blockage in the flow of (causal) information between the two points" in all probabilistic model that is compatible with the given DAG.

To be specific, let $\mathcal{D}$ be a DAG on a finite set $A$. Then the notion of $d$-separation is designed so that, for two distinct variables $x, z \in A$ and a set of variables $B \subseteq A$, the followings are equivalent:

1. $x$ and $z$ are $d$-separated by $B$;
2. $x$ and $z$ are conditionally independent, given $B$, for any probability distribution of variables in $A$ for which $\mathcal{D}$ is a Bayesian network.

(See Theorem 1.2.4 of Pearl (2009), for instance.)

In light of this, it should be clear that $a$ is not $d$-separated from itself by $B = \{b\}$ in your case. This also tells that we should not allow $x$ and $z$ to be the same in your definition.